The general formalism of some scattering processes involving photons

1 8.B.1

]

Nuclear Physics 87 (1966) 51--64; (~) North-Holland Publishing Co., Amsterdam Not to be-reproduced by photoprint o r

THE GENERAL FORMALISM

microfilm

without written permission from the publisher

OF SOME SCATTERING PROCESSES

INVOLVING PHOTONS G. RASCHE Institut fiir Theoretische Physik der Universit6t Ziirich

and W. S. WOOLCOCK Department of Physics, University College, London t Received 27 December 1965 Abstract: By correctly defining the polarization vectors of photons and the spinors for particles of spin ~-according to the prescription of Jacob and Wick, we show how to obtain the partial wave expansions of the invariant reaction amplitudes for the processes ~+N -~ ~ _ N , T--N -+ ~' ?N and 7-~-;e -+ N-~N. No specific representation of the Dirac matrices is necessary.

1. Introduction In an earlier paper 1) we gave a method for obtaining the partial wave expansions o f the invariant reaction amplitudes for the processes 7r + N ~ rc + N and ~ + 7r --+ N + N without using the low-energy representation o f the Dirac matrices. This m e t h o d will be extended in this paper to processes involving photons. It is possible in this way to obtain neatly and straightforwardly the connections between the invariant amplitudes and the partial wave expansions for the processes 7 + N --* ~r+ N, 7 + N ~ 7 + N and 7 + 7 --" N + N . This last process is of interest in analysing nucleon C o m p t o n scattering by dispersion relations. In sect. 2 we set up the necessary tools for our calculations, namely the construction o f the spinors and the polarization vectors for the nucleons and p h o t o n s respectively. Pion p h o t o p r o d u c t i o n f r o m nucleons is treated in sect. 3 and nucleon C o m p t o n scattering in sect. 4. in sect. 5 we consider the production of a nucleon-antinucleon pair by p h o t o n - p h o t o n collision. The appendix discusses the interpretation of the usual p h o t o p r o d u c t i o n analysis in terms o f electric and magnetic multipole radiation. Isospin is not considered, as it is not relevant to the purpose of the paper.

2. Definition of Spinors and Polarization Vectors We use here the same metric tensor and conventions as in ref. 1). The spinors u(r)(p) and v(O(p), corresponding to nucleons and antinucleons respectively, with t This work was supported in part by a grant from the Office of Aerospace Research (European Office) U.S.A.F. and by the Schweizerische Nationalfonds.

51

52

G. RASCHE AND

W.

S. W O O L C O C K

helicity ½r (r = ___l) and linear momentum p, have to be constructed according to the phase conventions of Jacob and Wick 2). This has been done explicitly in ref. 2) and we briefly quote the results here. Let k be the unit vector along the z-direction of some cartesian coordinate system in space, and let u~~ be the spinor for a particle at rest in that system, satisfying the equations :.

. (r)

oU~

~

(r)

~U~

~

~

, (r)

u3~

~

... (r)

~- tu~

-(r).

,

bl~

(s)

u~

~

t~rs.

We require further that

___io )4

= u?+L

Given any unit vector/3, with polar angles 0 and 4, the spinors u~~ satisfying the equation ( a . p)@') = ru~,") are defined by

u(•)

~

e-~(s-~)4'd~~ s z, r k(O~uf ~) ] e

(la)

u%

Ob)

E s

with the special case

•

Finally, the spinor u(r)(p) for a particle with linear momentum IPl in the direction 13 and helicity ½r is given by

u(')(p) where El,

=

pO

=

=

M-iV

•p

[2~f+-~)]~

@'),

(ic)

x/~++p2.

The spinors v(r)(p) for antiparticles are defined by the relations v},"~ = l"r 7 5 ~)(r)(p)

=

u ~( -

r),

M + iv " p

[2M~n;)~)]¢

(2a) vg ).

(2b)

For further details, see ref. ~). Actually, the rotation (la) and the pure Lorentz transformation (lb) are performed in the reverse order to that of ref: 2). This is justified, since a pure Lorentz transformation in the direction ~ with a certain velocity V, followed by a space rotation, which transforms ~ into p, is the same as the identical space rotation, followed by a pure Lorentz transformation in the direction/3 with the velocity V. We next have to define the polarization vectors of a photon with linear momentum in the direction k and helicity 2 ( = + 1). To do this we again consider a fixed cartesian coordinate system in space, with the unit vectors ~i (i = 1, 2, 3) pointing in the positive directions of the three axes. The polarization vectors corresponding to the two states with definite helicity of a photon with linear momentum in the direction of ~3

PHOTON SCATTERING

53

are then defined to be |

~")(~3) =

,~2 (2~, + ie:),

2 = ___1.

(3)

An important consequence of this definition is that the parity factor ~/of the photon becomes r/ = - 1 (see ref. 2)). The longitudinal polarization vector corresponding to the unphysical helicity 2 = 0 is evidently

~(o)(~3) = ~ . Given any unit vector/~ with polar angles O, 4), the polarization vectors e(z)(/~) corresponding to photons with linear momentum in the direction k and helicity 2 are constructed by the following three-dimensional rotation:

~")(r,) = Z e - " " - ~dL(O)~")(~) • #

Inserting the explicit forms of the functions d2x (see, for example, ref. 2)), we have e(°)(k) = sin 0 cos q3/~1+sin 0 sin (D{~2q-COS Oe 3 = ~,

da)(fO =

1 [2(cos2½0_e2,~.C,sinZ½0)~'

,/2

+ i(cos z 10 + eZU~sin: 10)~ 2 - 2 sin 0 eU~3], (4 = +__1).

(4)

The polarization vector e
3. Pion Photoproduction from Nucleons

We consider the process

~(k, 2 ) + N ( p l , rl) --, ~ ( q ) + N ( p 2 , r2). Using invariance under the inhomogeneous orthochronous Lorentz group, together with gauge invariance, the general form of the S-matrix element for this process, taken between plane wave states with a conventional normalization, is (see, for example, refs. 3, 4))

(q, P2r2[S]k2, plrl) M = -i(2n)46(4)(pl+k-p2-q)2(E~E2e~[kl) ~ T(q, p2r2; k2, plrl),

(6)

T(q, par2; k2, plrl) = ~t¢r2)(pz)(MaA+MBB+McC+MDD)u(")(px ).

(7)

where

54

G. R A S C H E A N D

W.

S. W O O L C O C K

Here A, B, C, D are scalar functions of any two independent kinematical invariants of the process, and co, Ikl, E1 and E 2 are the pion, photon, initial and final nucleon energies, respectively. Further, with e = e(;')(~:) = (0, ~(~)(k)), we have MA = --757 " ~;7 " k, MB = - 2 7 s ( P " eq' k - P . M c = i75(7" eq " k - 7

k q " e),

" kq" e),

M o = 2i~5(7 • EP. k - 7

• k P " e - i M 7 • 57" k),

with P = ½(p~ +P2). We specialize n o w to the centre-of-mass system, where k = -Pl,

q

-----

--P2"

The magnitude v(v') of the relative velocity of the two particles in the initial (final) state is given by i~ -

W E1

Wlql

i.)' -

E2(W-E2) '

'

where W = ] k [ + E t = c o + E 2 is the total energy in the centre-of-mass system. By comparing eq. (6) with the appropriate equations of ref. 2), we find that ../

f~2;a~( 0, qS) = - 4 nM- W ~ q-k T ( q , - q r 2 ;

k2, -krl),

(S)

where 0 and q5 are the polar angles of ~ in some system of spherical polar coordinates with k as the polar axis t We now reduce eq. (7) by introducing the explicit expression (lc) for u ( r ' ) ( - k ) , and the corresponding expression for ~(~2)(_q). Defining M1 -= i

tr

• g

k 2

,

M2 = ( ' " q ) ( ' " (g × ~)), kq kq

q "g

M , _ iaq2" q I q N1 =

" g--

k2

'

(q " k)(k" ~)1 k2

'

k(W+M)

2M(M + el) (M + E2) ' * See also ref. t), particularly the derivation of eq. (38). (This equation is in error by a factor 2.) Note also that from now on, except in the definition of vl, we use, q, k to denote lql, Ikl respectively.

PHOTON SCATTERING

55

a straightforward but very long calculation gives =

--q

2J

-ll,

~(.2)( _ q ) m , u('~)( - k) = N, q ~ ] ' [ ( m + E2)( W - m ) m 3 - ( W + m ) q m 4 ] u ~ ~, ~("'~)(-q)Mcu("')(-k) = NI -(r2)u_q-[(2MvlW_M-)(M + E2)Mt + (W +~M~) .(rD u-i,,

+(M+E2)qM3+q2j~I4

~(r2)(_ q)MD u(r,)( _ k) = N, ~ ~2q)[(M+ E2) ( W - M ) M l + (W + M)qMz]u~lk ) , with

vl =

q'k_ 2M

1 [(W_Ez)(W_E1)_lqllklcosO]. 2M

Substituting these results into eqs. (7) and (8), we get 4-

•

V t't=(r2)m u ('')~

(9a)

i=I

in which thefi are the following linear functions of A, B, C, D:

s, = N (M+E )IA + c2M f2 = N2q

[

,

2Mv, -A+

(W+M)

f3 = N2 ( M + Ez)q [( W - M ) B + ( C - D) ], f , = N2q2[ - ( W + M ) B + ( C - D ) ] .

(9b)

Here the constant =/ N2 =

M Vq N

k

We must now reduce the expression (9a) further. Using eqs. (la), (Ib) and (3) and eq. (17) of ref. ~), we find that f~2; a~,(O, ~b) =

t eii(2a_rl +r2)dp ,,/2

× [(2+ rm)(br,. -r~ COS10--r26 .... sin ½0)(--f~ +fz cos 0) +sin 0(3 .... cos ½0+r13~" -~2 sin ½0){(1 + 2 r ~ ) f z + 2 r z f 3 + 2 r z f 4 } ] . Use has been made of the fact that

(~, + rl )e-~i(rl +r2)4,

= (~, + r l ) e ~ i ( 2 ~ . - rl +r2),

since 2 + r 1 # 0 i f and only i f 2 - r 1 = 0.

(10)

56

G. RASCHE AND W. S. WOOLCOCK

To get the partial wave expansions of thefi, we first expandf,~; ~,~(0, ~) according to eq. (31) of ref. 2):

f,2;a,,(O, q5) =

1 1 r 2IT s (W)I2zI r j>dz-lr~.-~,~t~,je s r m ii(2x+,2-,1)÷ • +~)(~

1 ~(j

(11)

The parity relations (-½rEITs(W)l-2, -½q> = -(½r21TS(W)[2, ½rt> given in eq. (43) of ref. 2) reduce the number of independent reduced matrix elements to four for each value of J. Equating the right sides of eqs. (10) and (11), the q%dependence drops out and by giving the helicity indices special values we get four independent equations, which can be solved for the amplitudesfv For convenience we write down the partial wave expansions of the following combinations:

,,/2 ~ { - t I T t + l l 1 - ½ ) (f3 + f , ) = ~k(f3--f4) =

~2

~ (½[Tt+lll-½>

(f,+f2) = 7-~'

1

(P;'+~(x)-P;'(x)),

x//(/+2~) 1

x/~(e;'+,(x)+P;'(x)),

[ <½1T'+II1½>(P;+&)+P;(x)) e;(,O)l ,

( f l _ _ f 2 ) ~_~ ~

i

~/

[ <__1~.IT/+½ ]lg)(Pz+l(X)-P,(x)) 1 ' ' ,

with x = cos 0. We now expand the final states IJJ3½r> into states with definite orbital angular momentum l and parity ( - 1 ) t+l, which we label by [JJ31~]f(~ = +__1), with orbital angular momentum l~ = ( J - ½ c 0 and parity P = (-1)s+i~:

IdJ31r> =

1

~2(]JJ3

l+ 1]f+rlJJ3 I- 1If).

(13)

Further, we expand the initial states [JJ31-4- ½) into states [JJ3e]i (e = +__1) defined by [JJ3e]i :

IJJ31½>+g~lJJ31-1>,

with K+I = --tj+~,

K+IK_ l = -1.

(14)

PHOTON SCATTERING

57

Note that the states IJJ3g]i are not normalized and, further, that all these "ket" symbols have to be interpreted according to ref. 2) t. The equations inverse to (14) are

[JJ31-½} = ~/(d - ½)(J + ~) ([dd3 - (2J+i)

lJ J31½) --

1]i-

IJJ3 + 1]~),

1 [(J+~)lJJ3+l]i+(J-½)lJJ3-1]i ]. ( 2 J + 1)

(15)

Finally, it is customary to define four new amplitudes E~+, Ml+ by

f[JJ3 l+alTt+~lJa3+ 1]i = 2ix/qk(/+ 1)E,+, f[JJ3 l+ llTl+~[ J J3- 1 ]i = 2ix/qk(l+ 1)Ml+, f[JJ3 I-aITt-~[JJ3 - 1]i = - 2&/q-klE,_, f [JJ3 l_ ~IU-÷I J J3 + 1 ]i = 2ix/qklM,_.

(16)

The notation E, M will be discussed in detail in the appendix. Inserting (13), (15) and (16) into (12) and altering the summation variable where necessary, the familiar partial wave expansions 4) emerge:

f~ = -

• {P;+ l(X)(IMt+ +El+)+P;_ l(x)[(/+ 1)Mr_ +El-]}, l

f2 = -

y~ {(t + 1)M, + + 1M,_ )P;(x), l

(fa+f4) = -

E(P;+I(X)--P~'(x))(EI+--Mt+--M(t+,)---E(t+I)-), /

(f3 - f ~ ) = -

Z

(e;; ,(x)+ e;'(x))(<+ - M , +

+M.+._ +E,+._).

We also give here the relations between the four independent reduced matrix ele' '+½ ]1, ~rl) 1 ments (~rz[T and the amplitudes E{,+l)- El+, M(,+O_, M,+, since they are used in the literature *t. The relations are (½1T'+~It½} = N3[I(M,+ --E(l+ 1)-)+(l+2)(El+ +M(l+,)-)], (--½1Z'+~ll½) = N3[(l+2)(E,+ -M(,+ ,)_)+ I(M,+ +E(l +,)_)], (½1Tl+ffll--½) = N3x/I(I+2)[MI+-EI+-M(,+~)_-E(t+I)_], (-½1T'+~-[1--½} = N3~/l(l+ 2)[Ml+ -E,+ +M(,+,)_ +E(l+,)_], with

N3 = ix/lqk.

* Note particularly pages 410-411. What we are really doing is making a decomposition of the reduced matrix elements appearing in eq. (12). ** See, for example, the paper of Hearn and Leader ~). In their eq. (4.18), the sign in the last relation differs from our result; also an overall phase factor has been chosen in a different way.

58

G. RASCHE AND W, S. WOOLCOCK

4. Nucleon Compton Scattering

We consider the process 7(kl, 2x)+N(P~, rl) -~ )'(k2, 22)+N(pz, rz). Using invariance under the full inhomogeneous Lorentz group and gauge invariance, the general form of the S-matrix element for this process, taken between plane wave states with a conventional normalization, is (see refs. 3, 5)) p2r2[S-I[kl)q;

(k2)].2;

girl) M F(k222, p2r2; k ~ ) h , p~rl), 2(E~ Ez[k~[[kzJ) ~-

= _i(2n)46(4)(pl+k_p2_k2)

(17)

where F ( k 2 " ~ 2 , P2 r2 ; k l "~1, P l

rl)

=

--[i(r2)(p2)'g~vgl~VUvu(rD(P,),

6

(lS)

F u~ = Y', A i GI'~. i=1

The six quantities A i are scalar functions of any two independent kinematical invariants of the process, and Ea, Ik~[ (E2, [ka]) are the nucleon and photon energies in the initial (final) state. The four-vectors el = ~(~')(kl)= (0, dz')(/~t) ) and ez = d~2)(k2) = (0, da:)(~2) ) are the initial and final photon polarization vectors, respectively. Finally,

6~

p,up,~ -

p,2

NUN ~ ,

G~ ~ -

G~,~ = i y . KG~ ~,

N 2

N;'p '~_ N ~ p ' , ,

G7

G~ ~ = iT" KG~ ~,

=

(p,zN2)½

~,

G~ ~ = - ( N U(p,2N2)~ P ' ~ + N ~ P " ) 75(i7" K),

where the four-vectors are defined as follows: P = ½(P~ +P2), p'=p

K = 1(kl+k2), (PK 2K) K,

Q = (kl-k2),

N u = pflvPrrP' l( I')

Specializing to the centre-of-mass system, we have kl = - P t , Ikll = Ik21 = k,

k2 --- --P2,

E1

= E2 =

g.

The total energy is W = E + k , and the magnitude of the relative velocity of the two particles is the same in the initial and final states, with v = v' = W/E. By comparing eq. (17) with the appropriate equations of ref. 2), we find that M fz2r2; z,,,(O, (9) = -- - F ( k 2 22, - k2 r2 ; kl ,~1, 4nW

--

kl rl),

(19)

PHOTON SCAT'IERING

59

0 and q5 being the polar angles of/~2 in some system of spherical polar coordinates with/~1 as the polar axis. In the centre-of-mass system, the four-vectors p,u and N" become p,O =

_

W (1 + cos 0) ( 1 - cos 0)'

N O = O,

p,

W (~1-~" k2), k ( l - cos 0)

=

N = W(k 2×kl).

Substituting (19) into (18) and using the results of subsect. 2.1 of ref. 1) and the explicit forms (3) and (4) of the polarization vectors of the photons, a routine calculation gives

8nWfa2r2; am(0, ~b)e-/(~1-~'~-a2+~'2)¢ = 3,,r2 COS ½0[M(A 2 -- 2122 A1)- Wk(A5 - 2122 A4) - rl k W(21 --~~,2)A6]

+fir,,-r~ sin ½0[rlE(Az-2122A~)-rlMk(As-2122A4)+k()tz-2~)A3].

(20)

From eq. (31) of ref. 2) we have the partial wave expansion

=

1_E

,,

,

,

]ZFl>dkq_½r,,2 2 _ ½ r 2 ( 0 ) .

(21)

Conservation of parity implies that

< - 22, -½rzlTS ( W)[- 21, -½rl> = , and time-reversal invariance that

= <22, ½r21Ts(W)[21, ½rl>. There are therefore six independent reduced matrix elements for each value of d and six independent amplitudes fa2r2; )~m(0, qS). Following ref. 6) as closely as possible, we take these to be

01(O) = f l 1;1 l( 0, ~b),

02(0) = f - 1 - , ; I ,(0, qS)e-;*,

03(0) = f - l l ; 11(0 , q~)e-2i*,

04(0) = f l - x ; 11(0, ~b)el*,

05(0) = f l - 1 ; 1-1( 0 , ~b),

06(0) = f - m l ; 1-1( 0 , qS)e-314'-

The partial wave expansions of the O1(0) are given by the right side of eq. (21), with the special values of 21, rl, 22, r 2 inserted. Giving the appropriate special values to 21, rl, 22, r2 in eq. (20) and solving the six equations thus obtained for the Ai, one finds that A1 = k-

(Oa _ 2 0 3 +(25) - sinW½0 (06 + 2 0 4 - 0 2 )

,

60

G. RASCHE AND W. S. WOOLCOCK

A2

2n

k-

E

M 0 (Q1 + 2Q3-F~25)-W ( 0 6 -- 2ff~4 -- ~'22)1 , cos sin ½~

2nW

A3 =

k sin ½0

( 0 2 "1-Q6),

A, = 2rc [ c o ~ f l (~2t- 2f23 + f25)- - -M 0 (ff~6 "}-2 0 , - 02)] , sin cosE ½o( ~ + 2Q3 + Q5) -- sinM½~ (~6 -- 2Q4 -- ff]2)1 ,

As = 2n ~ A 6 --

2n

k cos ½0

(05 -- ~21).

The partial wave expansions of the A i are clearly very complicated. It is also possible to establish neatly by our method the connection between the As and the amplitudes G~ sometimes used in the literature 6, 7). The latter are defined by 6

f*2,2; J.lrl(0, ~) = ~ ~'~~ -(,2) G Hi U(")-~'l'

(22)

i=1

where H1 = g~" g l ,

HE = (~2" 81)(~1" g~),

H 3 = icr' (g, × / ~ ) ,

H 4 = i(gl" e~)~r • (~1 x/¢E),

u,

= i , . { ( 8 , . ~E)("* X ~ , ) - - ( ~

• ~1)("* X ~)},

U 6 = ia"" ((gl"/¢2)(g~ x ] ¢ 2 ) - ( g ~ ' / ¢ 1 ) ( 8 1 x ~¢l)}-

Putting p, = - k , and P2 = - k 2 in (18) and using eq. (lc), another long but by now familiar calculation gives ~(rE)[

b ~.* o k--Jn,2jG2vOllt

-

-

~uv,,(r,)[ ~J1 t~

1 --(r2) 2M u - G

!.

k--~ly

H 2

+ ( E - M)

H 3 --

COS 0 Hs + - H6 sin 2 0 sin E 0

-k~,

G~~ = cos OG~+ 1 [(E_M)H4_I_CIH1] '

G~ = ~

k E2H3

(l+c°s 0) (H~-H6)] , sin 20

G~~ = (W + M ) 6 7 + ! CE--(W + M ) G hE, 2M sin E 0 1 [CEH 1 +(E-M)(W+M)H,-], G~~ = cos OG~+ ~-M

G~, = ~kW I 2 H 3 + ( 1 sin - c o2sO0)(HsWH6) 1 "

(23)

PHOTON SCATTERING

61

To shorten the writing, we have omitted the polarization vectors and spinors from the last five equations. The constants C1, C2 are given by

C~ = ( E + M ) - ( E - M ) c o s

O,

-C2 = (E+M)(W-M)+(E-M)(W+M)

cos 0.

The following identity is used in making the reduction ( g l " / ~ 2 ) ( ~ " /~1)(kl x k2) = ( ~ / x g~) sin 2 0

-[(,,,.

r,~)(,,~ x ,L)-(,,~.

+ [(,,1 • ~)(,,~ x ~ ) - ( , ~ *

r,1)(,,1 x ~,~)] cos o • r,,)(,,, × ~,)].

I f we now insert eqs. (23) into eq. (18) and compare eqs. (19) and (22), we arrive at six equations giving the G~ as linear combinations of the A~. These relations are written in full in eq. (2.19) of ref. 6).

5. Nucleon Pair Production in Photon-Photon Collisions

N o w we consider the process 7(kl, A1)+7(kz, )]'2) -+ N(p2, r2)+N(Pl, rx). In this case, the general form of the S-matrix element is M

= - i(27z)46(4)(Pt +P2 -- ka - k2)

2(glE2Jklllk21) ~

U(p2r2, Pl r l ; k l '~I, k2)]-2), (24)

where U ( p 2 r 2 , p ~ r x . kt)].l ' k2)~2) =

(25a)

__G(r2)l', h0(22)/~, ~ " \/'2]°v k~'2)hpt~Vo(21){~. ~t ~# kn'l]"~,,(rl)(.~ v kP1],

6

;"~ = Z A, G; ~.

(25b)

i=1

The six (72~ are derived from the G~'~ by making the substitutions k2 ~

--k2,

Pl ~ --Pt

in the definitions of the four-vectors P, K, Q, P', N, and defining ½ ( M 4 - s u ) , where s -- - ( k l - p ~ ) 2 = - ( p z - k 2 ) 2,

u = -(-kl+p2)

(p'2N2)~ to

2 = -(-p~+k2)

be

2.

The six amplitudes A i are once again scalar functions of any two independent kinematical invariants of the process and, if the Mandelstam representation is assumed, they are the analytic continuations of the amplitudes Ai used in sect. 4.

62

G.

RASCHE AND

W.

S. W O O L C O C K

Going once more to the centre-of-mass system, we define k

:

k 1 :

P

--k2,

k = Ik[,

=

Pl

:

--P2,

p--[p],

E = w/MZ+p z,

W = 2E = 2k.

Comparing (24) with the appropriate equations of ref. 2) in the usual way, we have f'lr2;al;'2(O'q~) =

8~

k U(-pr2'pra;k21'-k22),

(26)

with O, • the polar angles of t3 in some system of spherical polar coordinates with as the polar axis. The kinematical relations in the centre-of-mass system are p,o = 0,

P' = p(cos O ~ - p ) ,

N O = O,

N = 2k(kxp),

(p'EN2)'~ = -2kZp 2 sin z O. We now make the usual reduction of the right side of eq. (25a), using the results of sect. 2 of ref. 1) and the polarization vectors given in (3) and (5). Another long calculation gives for the amplitude f,,r~;z~2(O, clO defined in (26) the following result:

- 1 6 n k ~ e-it~'l--Z2--~rl+X~r2)~¢ Jrlr2; p~

t'r~ cb)

),I).2~ ~ '

(27)

= p()q 22 A 1 + A2)6~,,2 + kr,(2, +).2)A3 ~rlr2 + k(2122 A4 + As)(M cos O6,,r2 + r 1 k sin 06~,. _r2)+ kP(22- 2~) sin OA6 dr,, -r~. The partial wave expansion of the left side of (27) can be written down immediately from eq. (31) of ref. 2). There are just six independent reduced matrix elements for each J, since

(½rl, ½r21Ts(W)[2122> = <-½rl, -½r2[rS ( w ) [ - 21, - 2 2 ) (conservation of parity) and

(½rl, ½r2[TJ(W)12122) = ( - 1)S(½r~, ½rzlTS(m)Wl,~z,h) (identical particles in the initial state). When 2 I, 22, rl, r2 take all possible values, there are thus only six independent functions of O on the left side of (27). Taking six suitable special choices of 2~, 22, r~, r2 we can define six amplitudes analogous to the f2i of sect. 4, and eq. (27) expresses these as linear combinations of the six Ai. These relations can then be inverted, in the same way as in sect. 4. One of us (W.S.W.) wishes to thank Professor W. Heitler for his kind hospitality at the Institut fiir Theoretische Physik der Universitfit Ziirich during parts of 1965.

PHOTON

63

SCATTERING

Appendix We show now that, in the static limit for the nucleon, the amplitudes El+, Ml+ defined in eq. (16) correspond to transitions induced by electric and magnetic multipole radiation, as suggested by the notation. We first recall that the final states [JJ31,]f defined in eq. (13) are eigenstates of parity: P[JJ3/~]f = (-)s+½~ [JJ3 ~Jr. Next, we expand the initital state l J J3, 2½r) in states of definite parity. To do this, •. .t -¢1_ we have first to decompose it into states ]jj32)lj J32r) corresponding to definite total angular momentum j, j', three-component J3, J3 and helicity 2, ½r of the photon and nucleon, respectively. Using eqs. (14) and (16) of ref. 2), the plane wave expansion into angular momentum eigenstates for a single particle, the Clebsch-Gordan series and the orthogonality relation (23a) of ref. 2), we have

[JJ3,2½r> = [4~z(2J+l)] -~

Z "f(2j+l)(2j'+l)(-1)3'-~C(jj'J;J3J;) jj, J3+J'3=.13 × C(JJ J,2, -~r)]Jj32)[JJ3~r ). • .t

.

1

""

.i .r l

(A.1)

The states rjj32) and ]j~½r) have now to be written as superpositions of states with definite parity. We denote these latter by [jj3a) and[j'j~Q (a, z = __+_1), where

PIJJ3~) =

( - lY+~-~IJJ3~),

(A.2)

Pl./'73 z) = ( - 1)J'-~'lj ' j ; z). Here a -- + 1( - 1) corresponds to electric (magnetic) multipole radiation. The relevant superpositions are

1 Ijj32) = ~ [[Jj31)+2lJj3--1)]' (A.3)

]j~'3½r) = ~

1

[fj~]; 1 ) + r [ j ~ / ; - 1)].

Substituting (A.3) into (A.1) and (A.1) into (14), we find that [JJ3e]i =

~ jj"

F(jj'J; J3 J'3).

J3 + J'3 = J3

[{C(jj'J; 1, -½)+ K,C(jj'J; 1, ½)}{[JJ31)lj~]; 1)+ Ij j3 - 1)[j~/~ 1)} + {C(jj'J; 1, - ½ ) - K~ C(jj'J; 1, ½)}{[JJ31)[j~/; - 1) +I JJ3 - 1)[j'j; - 1)}].

(1.4)

The precise form of F is not relevant to the argument which follows. All the states on the right side of (A.4) have definite parity, which can easily be read off from eq. (A.2). It is obvious from (A.4) that in general IJJ3e]i will contain for each parity both electric and magnetic multipoles. In other words, each of the matrix elements E~_+,

64

G.

RASCHE AND

W . S. W O O L C O C K

Mt+ defined in (16) will c o n t a i n n o n - z e r o c o n t r i b u t i o n s f r o m transitions i n d u c e d by both electric a n d m a g n e t i c m u l t i p o l e r a d i a t i o n . W e n o w go to the static limit in the d e s c r i p t i o n o f the nucleon. This can be d o n e b y r e s t r i c t i n g j ' to the value ½ a n d the p a r i t y o f the nucleon to + 1. T h e only nucleon 1 .t states c o n t r i b u t i n g to the right side o f (A.4) are then Izj31), J3. t ~ ± ½ . W h e n only these terms in the sum are taken, d e n o t e the resulting state by [JJ3e] s. O u r a p p r o x i m a t i o n is therefore to consider the m a t r i x elements r[JJ31~lTSlJJ3~] s instead o f

f [JJ3l~[ TS l JJ3g ]i. N o t e further that in the t r u n c a t e d s u m j is restricted to J__ ½, a n d t h a t

c(s +½~, ½, J; 1,-½) c(s + ½~, ½,s; 1, ½)

=

-K_p,

fl =

+_1.

W e therefore have

IJJ3E] s

Z

G(flJ;J3J'3)(K~-K-p)[IJ+½fl,J31)+lJ+½fl,j3-1)]l~J31).

(A.5)

3=+1 J3 + J ' 3 = J 3

Thus, for e fixed, only the t e r m in the sum for/~ = e does n o t vanish. T h e right side o f (A.5) is therefore the sum o f two states with p a r i t y ( - 1 ) s + ~ + ~ - ~ ( a = ___1). F o r the t r a n s i t i o n to a final state IJJ31~,]f, only the state with p a r i t y ( - 1 ) s + ~ contributes. Hence, if ~ = e, tr = + 1 a n d only the electric m u l t i p o l e state contributes, while if c~ = - e , a = - 1 a n d only the m a g n e t i c m u l t i p o l e state contributes. I t is this p r o p e r t y o f the limiting case o f a static nucleon which is expressed in the conv e n t i o n a l designation o f the a m p l i t u d e s defined in (16).

References 1) 2) 3) 4) 5) 6) 7)

G. Rasche and W. S. Woolcock, Nuclear Physics 68 (1965) 582 M. Jacob and G. C. Wick, Ann. of Phys. 7 (1959) 404 S. Gasiorowicz, Fortschr. d. Phys. 8 (1960) 665 G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1345 R. E. Prange, Phys. Rev. 110 (1958) 240 A. C. Hearn and E. Leader, Phys. Rev. 126 (1962) 789 M. Jacob and J. Mathews, Phys. Rev. 117 (1960) 854